Statistical Physics

1 Statistical Thermodynamics Professor Dmitry Garanin Statistical physics May 17, 2021 I. PREFACE Statistical physics considers systems of a large number of entities (particles) such as atoms, molecules, spins, etc. For these system it is impossible and even does not make sense to study the full microscopic dynamics. The only relevant information is, say, how many atoms have a particular energy, then one can calculate the observable thermodynamic values. That is, one has to know the distribution function of the particles over energies that defines the macroscopic properties. This gives the name statistical physics and defines the scope of this subject. The approach outlined above can be used both at and off equilibrium. The branch of physics studying non- equilibrium situations is called physical kinetics. In this course, we study only statistical physics at equilibrium. It turns out that at equilibrium the energy distribution function has an explicit general form and the only problem is to calculate the observables. The term statistical mechanics means the same as statistical physics. One can call it statistical thermodynamics as well. The formalism of statistical thermodynamics can be developed for both classical and quantum systems. The resulting energy distribution and calculating observables is simpler in the classical case. However, the very formulation of the method is more transparent within the quantum mechanical formalism. In addition, the absolute value of the entropy, including its correct value at T ! 0; can only be obtained in the quantum case. To avoid double work, we will consider only quantum statistical thermodynamics in this course, limiting ourselves to systems without interaction. The more general quantum results will recover their classical forms in the classical limit. II. MICROSTATES AND MACROSTATES From quantum mechanics follows that the states of the system do not change continuously (like in classical physics) but are quantized. There is a huge number of discrete quantum states i with corresponding energy values "i being the main parameter characterizing these states. In the absence of interaction, each particle has its own set of quantum states which it can occupy. For identical particles these sets of states are identical. One can think of boxes i into which particles are placed, Ni particles in the ith box. The particles can be distributed over the boxes in a number of different ways, corresponding to different microstates, in which the state i of each particle is specified. The information contained in the microstates is excessive, and the only meaningful information is provided by the numbers Ni that define the distribution of particles over their quantum states. These numbers Ni specify what in statistical physics is called macrostate. If these numbers are known, the energy and other quantities of the system can be found. It should be noted that also the statistical macrostate contains more information than the macroscopic physical quantities that follow from it, as a distribution contains more information than an average over it. Each macrostate k, specified by the numbers Ni, can be realized by a number wk of microstates, the so-called thermodynamic probability. The latter is typically a large number, unlike the usual probability that changes between 0 and 1. Redistributing the particles over the states i while keeping the same values of Ni generates different microstates within the same macrostate. The basic assumption of statistical mechanics is the equidistrubution over microstates. That is, each microstate within a macrostate is equally probable for occupation. Macrostates having a larger wk are more likely to be realized. As we will see, for large systems, thermodynamic probabilities of different macrostates vary in a wide range and there is the state with the largest value of w that wins over all other macrostates. If the number of quantum states i is finite, the total number of microstates can be written as X Ω ≡ wk; (1) k 2 the sum rule for thermodynamic probabilities. For an isolated system the number of particles N and the energy U are conserved, thus the numbers Ni satisfy the constraints X Ni = N; (2) i X Ni"i = U (3) i that limit the variety of the allowed macrostates k. III. TWO-STATE PARTICLES (COIN TOSSING) A tossed coin can land in two positions: Head up or tail up. Considering the coin as a particle, one can say that this particle has two \quantum" states, 1 corresponding to the head and 2 corresponding to the tail. If N coins are tossed, this can be considered as a system of N particles with two quantum states each. The microstates of the system are specified by the states occupied by each coin. As each coin has 2 states, there are total Ω = 2N (4) microstates. The macrostates of this system are defined by the numbers of particles in each state, N1 and N2: These two numbers satisfy the constraint condition (2), i.e., N1 + N2 = N: Thus one can take, say, N1 as the number k labeling macrostates. The number of microstates in one macrostate (that is, the number of different microstates that belong to the same macrostate) is given by the binomial distribution N! N wN1 = = : (5) N1!(N − N1)! N1 This formula can be derived as follows. We have to pick N1 particles to be in the state 1, all others will be in the state 2. How many ways are there to do this? The first \0" particle can be picked in N ways, the second one can be picked in N − 1 ways since we can choose of N − 1 particles only. The third \1" particle can be picked in N − 2 different ways etc. Thus one obtains the number of different ways to pick the particles is N! N × (N − 1) × (N − 2) × ::: × (N − N1 + 1) = ; (6) (N − N1)! where the factorial is defined by N! ≡ N × (N − 1) × ::: × 2 × 1; 0! = 1: (7) The recurrent definition of the factorial is N! = N (N − 1)!; 0! = 1: (8) The expression in Eq. (6) is not yet the thermodynamical probability wN1 because it contains multiple counting of the same microstates. The realizations, in which N1 \1" particles are picked in different orders, have been counted as different microstates, whereas they are the same microstate. To correct for the multiple counting, one has to divide by the number of permutations N1! of the N1 particles that yields Eq. (5). One can check that the condition (1) is satisfied, N N X X N! N wN1 = = 2 : (9) N1!(N − N1)! N1=0 N1=0 The thermodynamic probability wN1 has a maximum at N1 = N=2; half of the coins head and half of the coins tail. This macrostate is the most probable state. Indeed, as for an individual coin the probabilities to land head up and tail up are both equal to 0.5, this is what we expect. For large N the maximum of wN1 on N1 becomes sharp. To prove that N1 = N=2 is the maximum of wN1 ; one can rewrite Eq. (5) in terms of the new variable p = N1 −N=2 as N! w = : (10) N1 (N=2 + p)!(N=2 − p)! 3 FIG. 1: The binomial distribution for an ensemble of N two-state systems becomes narrow and peaked at p ≡ N1 − N=2 = 0. One can see that wp is symmetric around N1 = N=2; i.e., p = 0: Using Eq. (8), one obtains w (N=2)!(N=2)! N=2 N=2±1 = = < 1; (11) wN=2 (N=2 + 1)!(N=2 − 1)! N=2 + 1 one can see that N1 = N=2 is indeed the maximum of wN1 : The binomial distribution is shown in Fig. 1 for three different valus of N. As the argument, the variable p=pmax ≡ (N1 − N=2) = (N=2) is used so that one can put all the data on the same plot. One can see that in the limit of large N the binomial distribution becomes narrow and centered at p = 0, that is, N1 = N=2. This practically means that if the coin is tossed many times, significant deviations from the 50:50 relation between the numbers of heads and tails will be extremely rare. IV. STIRLING FORMULA AND THERMODYNAMIC PROBABILITY AT LARGE N Analysis of expressions with large factorials is simplified by the Stirling formula p N N N! =∼ 2πN : (12) e p In many important cases, the prefactor 2πN is irrelevant, as we will see below. With the Stirling formula substituted, Eq. (10) becomes p N ∼ 2πN (N=e) wN1 = p2π (N=2 + p) [(N=2 + p) =e]N=2+p p2π (N=2 − p) [(N=2 − p) =e]N=2−p r 2 1 N N = q N=2+p N=2−p πN 2p 2 (N=2 + p) (N=2 − p) 1 − N wN=2 = q ; (13) 2p 2 2p N=2+p 2p N=2−p 1 − N 1 + N 1 − N where r 2 w =∼ 2N (14) N=2 πN 4 is the maximal value of the thermodynamic probability. Eq. (13) can be expanded for jpj N: Since p enters both the bases and the exponents, one has to be careful and expand the logarithm of wN1 rather than wN1 itself. The square root term in Eq. (13) can be discarded as it gives a negligible contribution of order p2=N 2: One obtains N 2p N 2p ln w =∼ ln w − + p ln 1 + − − p ln 1 − N1 N=2 2 N 2 N " # " # N 2p 1 2p2 N 2p 1 2p2 =∼ ln w − + p − − − p − − N=2 2 N 2 N 2 N 2 N 2p2 p2 2p2 p2 =∼ ln w − p − + + p − + N=2 N N N N 2p2 = ln w − (15) N=2 N and thus 2p2 w =∼ w exp − : (16) N1 N=2 N p One can see that wN1 becomes very small if jpj & N that for large N does not violate the applicability condition jpj N.

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